On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions

Abstract

The vanishing viscosity limit is considered for the incompressible 2D Navier-Stokes equations in a bounded domain. Motivated by studies of turbulent flow we suppose Navier's friction condition in the tangential direction, i.e. the creation of a vorticity proportional to the tangential velocity. We prove the existence of the regular solutions for the Navier-Stokes equations with smooth compatible data and of the solutions with bounded vorticity for initial vorticity being only bounded. Finally, we establish a uniform L ∞ -bound for the vorticity and convergence to the incompressible 2D Euler equations in the inviscid limit.

Examples of Singular Limits in Hydrodynamics

“…Also, in [36], the vanishing viscosity limit is considered with the Navier (friction) boundary condition.…”

Eigenvalues of the Stokes operator versus the Dirichlet Laplacian in the plane

“…These boundary conditions, originally defined by Navier, have recently received considerable attention from fluid mechanics as a physically motivated replacement for Dirichlet boundary conditions, as they allow a thorough characterization of the boundary layer. See for instance [Clopeau et al 1998;Lopes Filho et al 2005;Kelliher 2006; Iftimie and Planas 2006;Iftimie and Sueur 2006]. We also discuss Neumann boundary conditions for the velocity and for the vorticity, and Robin boundary conditions for the vorticity.…”

Rate of Convergence in Inviscid Limit for 2D Navier-Stokes Equations with Navier Fricition Condition for Nonsmooth Initial Data

“…With the Navier friction condition, it is easier to handle the inviscid limit in a bounded domain. Indeed, they [7][8][9] showed that, under the Navier friction condition with a given friction coefficient, the solutions of the Navier-Stokes converge to those of the Euler equations as the viscosity tends to zero for various classes of initial velocity. Also, if the initial velocity is smooth enough, it is known that the rate of convergence is linearly proportional to the viscosity [10,11] .…”